I have been informed that on March 7th from 6:00am to 6:00pm Central Time Lamar University will be doing some maintenance to replace a faulty UPS component and to do this they will be completely powering down their data center.
Unfortunately, this means that the site will be down during this time. I apologize for any inconvenience this might cause.
Paul
February 18, 2026
Section 1.3 : Trig Functions
3. Determine the exact value of \(\displaystyle \sin \left( {\frac{{7\pi }}{4}} \right)\) without using a calculator.
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First we can notice that \(2\pi - \frac{\pi }{4} = \frac{{7\pi }}{4}\) and so the terminal line for \(\frac{{7\pi }}{4}\) will form an angle of \(\frac{\pi }{4}\) with the positive \(x\)-axis in the fourth quadrant and we’ll have the following unit circle for this problem.
The coordinates of the line representing \(\frac{{7\pi }}{4}\) will be the same as the coordinates of the line representing \(\frac{\pi }{4}\) except that the \(y\) coordinate will now be negative. So, our new coordinates will then be \(\left( {\frac{{\sqrt 2 }}{2}, - \frac{{\sqrt 2 }}{2}} \right)\) and so the answer is,
\[\sin \left( {\frac{{7\pi }}{4}} \right) = - \frac{{\sqrt 2 }}{2}\]